Finite Group Databasebeta

← #37534762

Subgroups — #37534762

Order 23168.

406 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153842C212yesmaximal, complemented
2#26041860C211yesmaximal, characteristic, complemented
3C2×#19153744C212yesmaximal, complemented
4#26041048C211yesmaximal, characteristic, complemented
5#390×C181C211yesmaximal, characteristic, complemented
6C2×#15668449C2212yes—
7#19153842—24no—
8#19153842—24no—
9C2×#15668437—22no—
10C2×#15668434—22no—
11C2×#15668433C2212yes—
12#19153796—22no—
13#19153791—22no—
14#19153790—22no—
15#19153744—24no—
16C22×D4×C181C2211yescharacteristic
17C2×C4⋊C4×C181C2211yescharacteristic
18C2×C22⋊C4×C181—22no—
19C2×C42×C181C2211yescharacteristic
20#15668449—22no—
21#15668449—22no—
22#15668449—22no—
23#15668449—22no—
24C2×Dic362D412yes—
25C2×D724—22no—
26C2×D724D412yes—
27C2×C22⋊D181—22no—
28C2×C4×D181—22no—
29#15668437—28no—
30#15668434—28no—
31#15668433—22no—
32#15668433D414yescomplemented
33C2×#13089678—22no—
34C4⋊Dic181—22no—
35#15668426—24no—
36C24×C181—21no—
37C2×D4×C181—22no—
38C2×D4×C181—22no—
39C2×D4×C181—21no—
40C2×D4×C181D412yescomplemented
41C22×C4×C181—21no—
42C22×C4×C181D412yes—
43C22×C4×C181C2311yescharacteristic
44C4⋊C4×C181—22no—
45C22⋊C4×C181—44no—
46C42×C181—22no—
47C22×D181—22no—
48Dic362—22no—
49Dic362—22no—
50D724—22no—
51D724—22no—
52D724—82no—
53C22⋊D181—42no—
54C22⋊D181—42no—
55C2×Dic181—22no—
56C4×D181—42no—
57C4×D181—42no—
58#13089678—24no—
59C23×C181—22no—
60C23×C181—42no—
61C23×C181—42no—
62C23×C181—41no—
63C23×C181—41no—
64C23×C181C2×D411yescharacteristic
65D4×C181—24no—
66D4×C181—42no—
67C2×C4×C181—22no—
68C2×C4×C181—22no—
69C2×C4×C181—41no—
70C2×C4×C181—41no—
71C2×C4×C181—42no—
72C2×C4×C181C2×D412yes—
73C2×C4×C181C2×D412yes—
74C2×C4×C181C2×D411yescharacteristic
75C2×C4×C181#4111yescharacteristic
76C22×C181—21no—
77C22×C181—21no—
78C22×C181—44no—
79C22×C181—42no—
80C22×C181—82no—
81C22×C181—81no—
82C22×C181—81no—
83C22×C181#12011yescharacteristic
84C22×C181#12111yescharacteristic
85C22×C181#12711yescharacteristic
86C2×D181—42no—
87C2×D181—42no—
88C2×D181—42no—
89C4×C181—42no—
90C4×C181—42no—
91C4×C181#12012yes—
92C4×C181#12112yes—
93Dic181—42no—
94C2×C181—21no—
95C2×C181—21no—
96C2×C181—82no—
97C2×C181#39211yescharacteristic
98C2×C181#45311yescharacteristic
99C2×C181#45411yescharacteristic
100D181—82no—
101C181#200611yescharacteristic, complemented, Sylow
102#2006—1811nomaximal, Sylow
103C2×AGL(1,ℤ/8)—1812no—
104#537—1811no—
105C2×#104—1812no—
106#390D18111yescharacteristic, complemented
107#343—1811no—
108C2×#41—1812no—
109C22×D4C2×D18111yescharacteristic
110AGL(1,ℤ/8)—3624no—
111AGL(1,ℤ/8)—3624no—
112C2×SD16—3622no—
113C2×D8—3622no—
114C2×M16—1812no—
115C22⋊Q8—3622no—
116#121—3622no—
117#121—3622no—
118C2×C4⋊C4C2×D18111yescharacteristic
119C2×C22⋊C4—22no—
120C2×C42C2×D18111yescharacteristic
121#104—3624no—
122C24—21no—
123#41—3622no—
124#41—3622no—
125#41—3622no—
126#41—3622no—
127C2×Q8—1812no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—21no—
131C2×D4—3622no—
132C2×D4—3622no—
133C2×D4—1812no—
134C2×D4D72412yescomplemented
135C22×C4—21no—
136C22×C4—3622no—
137C22×C4C22⋊D18112yes—
138C22×C4C22×D18111yescharacteristic
139SD16—3628no—
140D8—3628no—
141M16—3622no—
142M16—1814no—
143C2×C8—3622no—
144C4⋊C4—22no—
145C4⋊C4—3622no—
146C22⋊C4—44no—
147C22⋊C4—3624no—
148C42—22no—
149C23—22no—
150C23—42no—
151C23—42no—
152C23—41no—
153C23—41no—
154C23—3622no—
155C23C2×C22⋊D18111yescharacteristic
156Q8—3622no—
157Q8—3622no—
158D4—24no—
159D4—42no—
160D4—3622no—
161D4—3622no—
162D4—14482no—
163D4—7242no—
164D4—7242no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—3622no—
171C2×C4—7242no—
172C2×C4—7242no—
173C2×C4D4×D18112yes—
174C2×C4D4×D18112yes—
175C2×C4C2×D72411yescharacteristic
176C2×C4#1566844911yescharacteristic
177C8—3624no—
178C22—21no—
179C22—21no—
180C22—44no—
181C22—42no—
182C22—82no—
183C22—81no—
184C22—81no—
185C22—7242no—
186C22—7242no—
187C22—7242no—
188C22#1915378411yescharacteristic
189C22#1915379011yescharacteristic
190C22#1915381211yescharacteristic
191C4—42no—
192C4—42no—
193C4—7242no—
194C4#1915378612yes—
195C4#1915378912yes—
196C2—21no—
197C2—21no—
198C2—82no—
199C2—14482no—
200C2#2604118811yescharacteristic
201C2#2604144611yescharacteristic
202C2#2604145111yescharacteristic
subgroupG/H
centerC22#19153784
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#390×C181C2
socleC22×C181#120
hypercenter#390D181
hypocenterC181#2006
LCSG ▷ C22×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#19153842C2semidirect
2. #26041860C2semidirect
3. C2×#19153744C2semidirect
4. #26041048C2semidirect
5. #390×C181C2semidirect
32. #15668433D4semidirect
40. C2×D4×C181D4semidirect
101. C181#2006semidirect
106. #390D181semidirect
134. C2×D4D724semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#191538422
#260418602
C2×#191537442
#260410482
#390×C1812
#2006181